A Graph-Based Hyper-Heuristic for Educational Timetabling ...eprints.nottingham.ac.uk/346/1/rxqEJOR.pdf · In press. European Journal of Operational Research, 2006 1 A Graph-Based

In press. European Journal of Operational Research, 2006

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A Graph-Based Hyper-Heuristic for Educational

Timetabling Problems

EDMUND K BURKE1, BARRY MCCOLLUM2, AMNON MEISELS3, SANJA PETROVIC1, RONG QU1*

1Automated Scheduling Optimization & Planning Group, University of Nottingham, Nottingham, NG8 1BB, UK

2School of Computer Science, Queen's University Belfast, Belfast BT7 1NN, UK

3Department of Computer Science, Ben-Gurion University, Beer-Sheva 84 105, ISRAEL

*corresponding author, email: [email protected], telephone: ++44 115 9514215, fax: ++44 115 8467591

Key words: heuristics, graph heuristics, hyper-heuristics, tabu search, timetabling

Abstract

This paper presents an investigation of a simple generic hyper-heuristic approach upon a set of

widely used constructive heuristics (graph coloring heuristics) in timetabling. Within the hyper-

heuristic framework, a Tabu Search approach is employed to search for permutations of graph

heuristics which are used for constructing timetables in exam and course timetabling problems. This

underpins a multi-stage hyper-heuristic where the Tabu Search employs permutations upon a

different number of graph heuristics in two stages. We study this graph-based hyper-heuristic

approach within the context of exploring fundamental issues concerning the search space of the

hyper-heuristic (the heuristic space) and the solution space. Such issues have not been addressed in

other hyper-heuristic research. These approaches are tested on both exam and course benchmark

timetabling problems and are compared with the fine-tuned bespoke state-of-the-art approaches. The

results are within the range of the best results reported in the literature. The approach described here

represents a significantly more generally applicable approach than the current state of the art in the

literature. Future work will extend this hyper-heuristic framework by employing methodologies

which are applicable on a wider range of timetabling and scheduling problems.


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1. Introduction

1.1 Timetabling Problems

Timetabling has attracted the attention of the Operational Research and Artificial Intelligence

research communities for more than 40 years. The general timetabling problem comes in

many different guises including nurse rostering (e.g. Cheang et al, 2003, Burke et al. 2004),

sports timetabling (e.g. Easton, Nemhauser and Trick, 2004), transportation timetabling (e.g.

Kwan, 2004) and university timetabling (Schaerf, 1999, Burke and Petrovic, 2002, Petrovic

and Burke, 2004), etc. Further examples of papers dealing with these kinds of problems can

be found in (Burke and Ross, 1996, Burke and Carter, 1998, Burke and Erben, 2000, Burke

and De Causmaecker, 2002, Burke and Trick, 2004). Perhaps the most widely studied class of

timetabling problem is that of educational timetabling. A wide variety of papers describing a

broad spectrum of educational timetabling methodologies has appeared in the literature over

the years. Overviews of the literature can be found in the following papers (Carter and

Laporte, 1996&1998, Bardadym, 1996; Burke, Jackson et al, 1997, Schaerf, 1999, Burke and

Petrovic, 2002, Petrovic and Burke, 2004).

In a general educational timetabling problem, a set of events (e.g. courses and exams, etc)

need to be assigned into a certain number of timeslots (time periods) subject to a set of

constraints, which often makes the problem very hard to solve in real-world circumstances.

Constraints can usually be divided into two types:

• Hard constraints have to be satisfied under any circumstances. For example, in exam

timetabling, two exams with common students involved cannot be scheduled into the

same timeslot. Timetables with no violations of hard constraints are called feasible

solutions.

• Soft constraints need to be satisfied as much as possible. For example, in exam

timetabling, exams taken by common students often need to be spread out over the

timeslots so that students do not have to sit in two exams that are too close to each

other. Due to the complexity of the real-world timetabling problem, the soft constraints


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may need to be relaxed since it is not usually possible to generate solutions without

violating some of them. Soft constraints are usually within the cost evaluation function

to evaluate how good the solutions are.

1.2 Approaches and Techniques in Timetabling Problems

The early days of research in educational timetabling investigated such approaches as graph

heuristics (see de Werra, 1985, Burke, Kingston and de Werra, 2003) and integer linear

programming (see Carter, 1986). Some of the early techniques are either impractical or too

simple to solve complex or large timetabling problems. Constraint based techniques have

been employed over the years to address timetabling problems (e.g. Deris et al, 1997, Banks,

Beek and Meisels, 1998, Nonobe and Ibaraki, 1998). Recently, meta-heuristic search

techniques (see Glover and Kochenberger, 2003) have been investigated and seem to have

been very successful in solving a variety of timetabling problems. These include Tabu Search

(e.g. Costa, 1994, Di Gaspero and Schaerf, 2000), Simulated Annealing (e.g. Dowsland, 1998,

Abramson, Krishnamoorthy and Dang, 1999) and Evolutionary Algorithms (e.g. Burke,

Newall and Weare, 1996&1998, Burke and Newall, 1999, Erben, 2000, Lewis and Paechter,

2004, Côté, Wong and Sabourin, 2005). Other new approaches and methodologies for

timetabling problems have also been studied as more problem solving experience is collected

and new technologies provide new breakthroughs. These include Case-Based Reasoning

(Leake, 1996) on educational timetabling (Burke, MacCarthy et al. 2000, 2001, 2003&2005,

Burke, Petrovic and Qu, 2006) and on nurse rostering (Beddoe and Petrovic, 2005), fuzzy

methodology on exam timetabling (Asmuni, Burke and Garibaldi, 2004), and hyper-heuristics

on timetabling (Burke, Kendall and Soubeiga, 2003, Gaw, Rattadilok and Kwan, 2004, Burke,

Dror et al, 2005, Burke, Petrovic and Qu, 2006, Qu and Burke, 2005).

The present work concerns educational timetabling including both exam and course

timetabling problems (Carter and Laporte, 1996, Carter and Laporte, 1998). The state-of-the-

art approaches in educational timetabling usually employ specially tailored heuristic/meta-

heuristic approaches (e.g. Carter, Laporte and Lee, 1996, Di Gaspero and Schaerf, 2000,


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Caramia, Dell’Olmo, and Italiano, 2001, Burke and Newall, 2002, Merlot et al, 2002, Socha,

Knowles and Sampels, 2002, Asmuni, Burke and Garibaldi, 2004, Abdullah et al, 2004,

Burke, Bykov et al, 2004). These approaches invest a significant amount of development

effort in the production of “fine tuned” techniques that are specially built for the particular

problems in hand. The objective of the present paper is to develop an approach which is more

widely applicable and fundamentally more general than the approaches mentioned above. Our

goal is not to “beat” them but to obtain comparable results by only employing general and

simple techniques which can be applied to a wider range of scheduling problems.

1.3 Hyper-heuristics on Scheduling and Timetabling Problems

The development of hyper-heuristics is motivated by the goal of aiming at an increased level

of generality for automatically solving a range of problems (see Burke, Kendall et al. 2003).

A hyper-heuristic can be seen as an algorithm (on a higher level) which “picks” appropriate

heuristics (at a lower level) to be applied to the problems in hand. A hyper-heuristic is

concerned with the exploration of a search space of heuristics instead of dealing directly with

solutions to the problem. In hyper-heuristic research on timetabling and scheduling, different

techniques have been investigated as the low level and high level search strategies in solving

the problems. We can categorize the work in hyper-heuristics (in terms of the “low level”

heuristics employed) into two groups: improvement techniques and constructive techniques.

1.3.1 Improvement Techniques within Hyper-heuristics

In a hyper-heuristic, move strategies are usually employed as the low level heuristics to search

for solutions to timetabling and scheduling problems. Kendall, Cowling and Soubeiga (2002)

employed choice functions by which appropriate low level heuristics (moving strategies) can

be chosen and combined adaptively to search the solutions. Good results on project

presentation problems were presented compared with the solutions generated by a random

hyper-heuristic. A distributed choice function method was proposed by Gaw, Rattadilok and

Kwan (2004) within a hyper-heuristic for timetabling and scheduling problems.


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Burke, Kendall and Soubeiga (2003) employed a Tabu Search as the high level heuristic to

search through a space of moving strategies for university course timetabling and nurse

rostering problems. Good results on both of the problems tested indicated the generality and

efficiency of the hyper-heuristic approach. This approach is adopted and extended in (Burke,

Landa Silva and Soubeiga 2005) with the aim of investigating the learning of low level

heuristics that are suitable and effective for individual objectives in multiple-objective space

allocation and course timetabling problems. Promising results are obtained compared with the

state-of-the-art approaches. A genetic algorithm methodology was employed as the high level

approach by Han and Kendall (2003) and was tested on simulated course scheduling and

student project presentation problems. Dowsland, Soubeiga and Burke (2005) introduced the

Tabu Search hyper-heuristic, which was investigated in (Burke, Kendall and Soubeiga 2003),

within a Simulated Annealing in search of a set of low level heuristics (both neighborhood

structures and sampling policies within the solution space) to determine the shipper sizes in

transportation problems.

Burke, Petrovic and Qu (2006) employed a Case-Based Reasoning methodology as a

heuristic selector for solving course timetabling problems. Problem information is modeled

with the corresponding good meta-heuristics and stored in the Case-Based Reasoning system.

The new problems are solved by using the suggested meta-heuristics which worked well on

solving previous similar problems.

1.3.2 Constructive Techniques within Hyper-heuristics

There are only a few papers which employ constructive techniques as low level heuristics

within hyper-heuristics for timetabling and scheduling problems. Terashima-Marin, Ross and

Valenzuela-Rendon (1999) investigated employing Genetic Algorithms to evolve the

configurations of constraint satisfaction methods on constructing problem solutions. The non-

direct representations were suggested for Genetic Algorithms to solve large and complex

exam timetabling problems.

Ross et al (2003) investigated a genetic-based hyper-heuristic employing 4 basic


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constructive heuristics on one dimensional bin packing problems. Optimal or near optimal

results have been found and potential research issues were also discussed.

Another technique was studied by Asmuni, Burke and Garibaldi (2004) who employed

fuzzy rules to combine graph heuristics to construct exam timetables.

Burke, Petrovic and Qu (2006) employed Case-Based Reasoning as a heuristic selector for

solving exam timetabling problems. Problem solving situations and the corresponding

constructive heuristics were stored in the Case-Based Reasoning system. Solutions for new

problems were constructed by repeatedly using the constructive heuristics suggested by the

system. Benchmark exam timetabling problems were tested and the results were within the

range of those generated by using the state-of-the-art approaches.

The graph based hyper-heuristic (GHH) described in this paper is a constructive approach that

employs different graph heuristics during the process of constructing the solution according to

the different problem solving situations that might occur at particular times. The next section

presents the GHH approach upon graph heuristics and a random ordering method. Some

fundamental issues concerning the search space and solution space are also addressed. The

investigation and experimental study of GHH and of a multi-stage GHH that are developed

for both exam and course timetabling problems are presented in section 3. Our conclusions

and potential extensions of this work are presented in the final section.

2. The Graph Based Hyper-heuristic (GHH) Approach

2.1 Graph Heuristics

Graph heuristics are widely studied methods which were developed during the early days of

research on timetabling problems (e.g. Welsh and Powell, 1967, Brelaz, 1979). For an

introduction to such approaches see (Burke, Kingston and de Werra 2004). They are used in

sequential (or constructive) solution methods to order the events that are not yet scheduled

according to the difficulties of scheduling them into a feasible timeslot (without violating any


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hard constraints). The difficulties are represented by the degrees of the vertices in the graph,

which model the timetabling problem by representing the events as vertices and conflicts by

edges. We say two events in timetabling problems have a conflict if they involve the same

students. The difficulty of an event is represented by the number of conflicts it has with the

others. The objective is to construct a timetable by scheduling the most conflicting events one

by one into feasible timeslots, satisfying as many of the soft constraints as possible.

Within our graph-based hyper-heuristic, we employ graph heuristics to schedule a number

of events at each step during the construction of the solution. The work presented in this paper

investigates the following 5 low level heuristics, with and without randomness (introduced by

a random ordering method):

• least Saturation Degree first (SD). Events are ordered (in an increasing order) in terms

of the number of feasible timeslots available in the partial solution at that time. The

priorities (degrees) of events to be ordered and scheduled are changed dynamically as

the solution is constructed.

• largest Color Degree first (CD). Events are ordered (in a decreasing order) in terms of

the number of conflicts (events with common students involved) that they have with

those already scheduled in the timetable. The degrees of the events not yet scheduled

are changed according to the situations encountered at each step of the solution

construction.

• Largest Degree first (LD). Events are ordered decreasingly by the number of conflicts

they have with other events. This heuristic aims to schedule first those events which

have the most conflicts.

• Largest Enrollment first (LE). Events are ordered (in a decreasing order) by the number

of students enrolled. This heuristic schedules first those events with the largest numbers

of students.

• Largest Weighted Degree first (LWD). Events are ordered (in a decreasing order) by

the number of conflicts, each of which is weighted by the number of students involved.


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Among events with the same degree, this heuristic gives higher priority to those with a

larger number of students involved.

• Random Ordering (RO). Events that are not yet scheduled are ordered randomly. This

basic heuristic brings some randomness into the scheduling, which sometimes produces

better results in timetabling problems as it increases the exploration of the search space.

2.2 The Graph Based Hyper-heuristic Framework

In the hyper-heuristic framework that we present in this paper, Tabu Search is employed to

search for the list of low level heuristics (permutations of graph heuristics and a random

ordering method), which are used to construct the complete solutions for timetabling

problems. Tabu Search has been widely employed for solving complex scheduling and

optimization problems (Glover and Kochenberger, 2003). The basic approach searches the

search space of the solutions by iteratively moving to the best neighborhoods of the current

solution, whilst keeping a record of recently visited solutions which it cannot re-visit again for

a certain number of steps (known as the tabu tenure).

Figure 1 presents the pseudo-code of the Tabu Search within the hyper-heuristic approach

we have developed. The search space of the Tabu Search (as the high level heuristic) consists

of all of the possible permutations of the low level heuristics described in section 2.1. A move

in Tabu Search is to pick a new heuristic list that is obtained by randomly changing two of the

heuristics in the previous heuristic list. The newly visited heuristic lists are added into the tabu

list (which has a length of 9. i.e. the tabu tenure is 9). The determination of this value is based

on our experiments and upon the value recommended from the literature (Reeves, 1996). of

course, what this means is that when the Tabu Search selects a heuristic list, that list cannot be

re-visited within 9 steps of the search. The objective of the Tabu Search is to find the heuristic

list that generates the best quality solution for the timetabling problem under consideration.

The search process of Tabu Search within the hyper-heuristic approach stops after a pre-

defined number of iterations (i in Figure 1), which we adapt according to the problem size.

We set it as 5 times the number of events to keep the computational time low. Of course it is


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possible to perform more iterations if computational time is not a key issue in the problem

being solved.

Each heuristic list selected by a move of the Tabu Search is used to construct a complete

solution, whose penalty is fed back to the Tabu Search as a guide to the search in the

following stages. As each heuristic in the heuristic list is employed to schedule a number (e in

Figure 1) of events into the timetable (the reason for doing this is explained below), the length

of the heuristic list (k in Figure 1) is then set as (number of events)/e.

INSERT FIGURE 1 SOMEWHERE HERE.

To reduce the size of the search space of the hyper-heuristic and thus reduce its computational

time, we added three mechanisms:

1) As the low level heuristics are graph heuristics which are used to construct the

solutions, a list of them may not guarantee to generate a feasible solution (see more

detail in Section 2.3). To reduce the time spent on implementing heuristic lists that

generate unfeasible solutions, we construct a ‘failed list’, which represents a flat

memory, to store the parts of heuristic lists that generate unfeasible schedules during the

solution construction (in the hyper-heuristic approach). The initial “failed list” is set as

empty and is updated after each step of the Tabu Search to store any new heuristic lists

that cannot generate feasible solutions. When a new heuristic list is selected by a move

of Tabu Search in the hyper-heuristic, it will be checked (before it is applied to

construct the solution) to see if it matches those stored in the ‘failed list’ that led to

unfeasible schedules. For example, if a part of the heuristic list ‘h1h2h3..’ (where h1,

h2 and h3 are low level heuristics) is stored in the ‘failed list’ because h3 cannot

generate a feasible schedule at that step, all of the heuristic lists selected later such as

‘h1h2h3h4 ..’ or ‘h1h2h3h5..’ can be ignored before being applied to construct a

solution. This mechanism cuts away a large section of the search space by ignoring the


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non-valid heuristic lists before applying them to solve the problems and reduces

significantly the computational time of the hyper-heuristic.

2) At each step of the solution construction employing a list of heuristics, a number of

events (set to 2 in our experiments because our initial testing indicated that this was an

appropriate value) are scheduled by the current heuristic in the list (rather than

scheduling just one event with the heuristic). This is motivated by the observation by

Burke, Petrovic and Qu (2006) that, when scheduling one event by a heuristic at each

step of the solution construction, the heuristics in the best heuristic list found by the

hyper-heuristic stay the same after a certain number of steps of scheduling. That is, the

best heuristic lists found tend to appear in the form of ‘h2h2h2…h2h1h1… h1…’.

Scheduling a number of events at each step reduces the size of the search space of the

hyper-heuristic without losing much quality in the solutions generated.

3) The initial heuristic list of the hyper-heuristic approach contains only the Saturation

Degree heuristic, in order to have a good starting point in the hyper-heuristic. This is

because the Saturation Degree heuristic orders the events not yet scheduled according

to the number of available timeslots, which changes dynamically during the search. It is

potentially (and experimentally tested to be) more effective (more often) than the other

heuristics that order the events in static lists. It is expected that the density of the

appearance of Saturation Degree in the best heuristic list will be higher than that of the

other low level heuristics.

2.3 Fundamental Issues within the Hyper-heuristic Approach

Hyper-heuristics have been attracting recent attention in timetabling research (see Burke and

Petrovic, 2002, Burke, Kendall et al. 2003, Petrovic and Burke, 2004). However, some

fundamental underlying issues have not been addressed in depth in the literature. Before

presenting the analysis of our experiments, we would like to discuss some of these issues.

As mentioned before, hyper-heuristics operate at a higher level than most meta-heuristic

approaches in the literature. They operate on heuristics rather than directly on the solutions by


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indirectly choosing a certain low level heuristic which then operates on the events (which

either moves the events in the timetable if the low level heuristics are moving strategies, or

assigns the events chosen to timeslots if the low level heuristics are constructive heuristics –

which is the approach studied here). Thus, it is necessary to distinguish between the search

space of the hyper-heuristic and the solution space of the problem.

Figure 2 presents the relationship between the search space of hyper-heuristics and the

solution space of the problem. Note that the search space of the hyper-heuristic (on the left

side of Figure 2) represents a space of heuristics and the solution space of the problem (on the

right side of Figure 2) represents a space of potential actual solutions (timetables). The

heuristics in the search space of the hyper-heuristic correspond to certain solutions of the

problem. Of course, the heuristics (which are constructive heuristics studied here) selected by

a move in the high level searching method may not guarantee the construction of a feasible

solution. This is because the moves in the hyper-heuristic search concern the change of

heuristics and not the actual assignment of the events. For example, in Figure 2 we can see

that heuristic A moves to heuristic B, or heuristic A moves to heuristic C (within the search

space of hyper-heuristic). The corresponding solutions (b or c in the solution space that are

constructed by B or C) in the solution space are not guaranteed to be feasible. The search

space of the hyper-heuristic (which consists of permutations of heuristics) is very large, with a

large number of heuristics that generate unfeasible solutions in the solution space of the

problem. In the figure, A and C generate feasible solutions, while B generates an infeasible

solution.


In Figure 2 solutions a and b (which correspond to heuristics A and B that are in the same

neighborhood in the search space of the heuristics), may not be in the same neighborhoods in

the search space of solutions. This gives the hyper-heuristic the ability of jumping (not

moving) within the problem solution space by moving among the neighborhoods defined at a


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higher level of search in the heuristic search space. In Figure 2, neighborhood moves are

represented as solid arrows and jumping moves are represented as dashed arrows. The

correspondences between the heuristics and actual solutions are represented by dotted lines.

Also note that in Figure 2, the solution space of the problem consists of all of the possible

solutions that may be obtained by neighborhoods moves, while they may not correspond to

any heuristic list in the search space of the hyper-heuristic. For example solution d may be

within the neighborhood of solution a in the search space of solutions. However d may not

have a corresponding heuristic in the search space of heuristics. Based upon the above

observations, we propose a hypothesis here: the hyper-heuristic (which operates at a higher

level of problem solving - solving the problems indirectly) may not be able to reach all of the

solutions in the solution space of the problem.

A deepest descent local search method is employed within the hyper-heuristic after each

move in the high level search. The deepest descent search tries to move the events to other

timeslots in the timetable that is generated by a heuristic (searched for by the high level

heuristic), aiming at improving the quality of the timetable as quickly as possible. This

process terminates as soon as no events can be moved to improve the timetable. The high

level heuristic will then make a move to another heuristic, which is used to construct another

timetable. An illustrative example is presented in Figure 3 in conjunction with the example

shown in Figure 2, where solution d in the search space of actual solutions might not

corresponds to any heuristic in the search space of heuristic lists, while it might be in a

neighborhood of solution a and thus may be visited by the deepest descent local search.


The deepest descent local search method is a simple but robust method, which does not

introduce extra domain knowledge within the hyper-heuristic framework. The motivation for

this is twofold: Firstly, the deepest descent in the search space of solutions tries to exploit the


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local areas (bringing the solutions obtained to their local optimum quickly), whilst the high

level heuristic in the hyper-heuristic tries to explore the search space; Secondly, the GHH thus

is potentially able to explore more of the possible solutions for the problem.

3. GHH on Exam and Course Timetabling Problems

3.1 Real-World Exam Timetabling Problems

We investigate our hyper-heuristic approach upon a set of graph heuristics by applying it to

the benchmark exam timetabling problems that are presented in (Carter, Laporte and Lee,

1996). These are real-world problems that have been tested by many state-of-the-art

approaches. The size of the problems ranges from 81 to 682 exams and from 611 to 18419

students. The density of the conflict matrix, which gives the ratio of the number of conflicting

exams over the overall number of exams, ranges from 0.06 to 0.42. The characteristics of the

problems are presented in Table 1.

INSERT TABLE 1 SOMEWHERE HERE.

The hard constraints considered in these problems are represented by the “conflicts” of

scheduling two exams with common students into the same timeslot. The soft constraint is

concerned with spreading out the students’ exams over the timetable so that students will not

have to sit exams that are too close to each other. The objective, which is the same as that

presented in (Carter, Laporte and Lee, 1996), is to schedule all of the exams into the timeslots,

while minimizing the cost on the violations of the soft constraint per student. The objective

function of GHH which calculates the cost of violations C(t) within solution t is presented in

formula (1) below:

C (t) = ( ∑=

4

0ssw × Ns) / S (1)

where


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ws = 2s, s = 0, 1, 2, 3, 4, is the weight that represents the importance of scheduling exams with

common students either 4, 3, 2, 1, or 0 timeslots away in timetable t.

Ns, s = 0, 1, 2, 3, 4, is the number of students involved in the violation of the soft constraint.

S is the number of students in the problem.

The lower the cost, C(t), the better the timetable is. We do not consider any infeasible

solutions (i.e. those with violations of hard constraints).

3.1.1 A Graph Based Hyper-heuristic upon a Different Number of Low Level Heuristics

We investigate here the effect of different low level heuristics on the behavior of the GHH for

exam timetabling problems. This helps us to gain a deeper understanding about general

hyper-heuristics which are applicable for a wider range of problems. The combinations of low

level heuristics employed are based on the Saturation Degree heuristic with a different

number of other heuristics with and without randomness. For all of the 11 problems presented

in (Carter, Laporte and Lee, 1996), 3 runs with distinct seeds are carried out and the costs of

the best solutions are presented in Table 2.


The values shown under the column “SL” and “SLR”, “SCL” and “SCLR”, and “SCLx” and

“SCLxR” in Table 2 are the costs of solutions obtained by GHH upon two, three and all graph

heuristics, with and without Random Ordering.

We assume that the larger the number of low level heuristics employed in the GHH, the

larger the size of the search space (of the GHH) will be. This implies that it is possible to

indirectly explore a larger part of the solution space of the problem (possibly containing more

and better results). For example, the search space of GHH upon SCLxR should include the

search space of GHH upon SCLR. This is apparent from Table 2, where the best results (in

bold) are mostly obtained by GHH with a larger number of low level heuristics.

Although employing a larger number of low level heuristics in GHH tends to obtain a


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better performance, this is not always the case. For example, GHH upon “SCLx” outperforms

“SCLxR”, which employs a larger number of low level heuristics, on 10 out of 11 problems.

The reason that GHH, when employing a smaller number of low level heuristics, sometimes

outperforms the approach with a larger number of low level heuristics is precisely because the

latter has a much larger search space. Some parts of the larger search space may not be

explored within the same search time (i.e. number of iterations). To clarify this issue we

carried out another set of experiments on “SCLxR” with a higher number of iterations (10 *

number of events). The results obtained are presented in the last column entitled

“SCLxR(*10)” in Table 2. We can observe that GHH upon “SCLxR” obtained better results

(on 5 out of 11 problems than that of “SCLx”) by being given more searching time. As the

search space of GHH upon “SCLx” is a subset of that of GHH upon “SCLxR”, it is possible

to say that, given enough search time, the results obtained by GHH upon “SCLxR” will be at

least not worse than that of GHH upon “SCLx”.

We can also see that GHH upon “SL” and “SCL” with random ordering outperforms the

same GHH without randomness, which is not the case for GHH upon “SCLx”. When more

searching time is given, GHH upon “SCLxR” performs much better, meaning that GHH

employing heuristics with random ordering performs better than without randomness. We

may then conclude, from the above observations, that the larger the number of low level

heuristics in GHH, the better it may perform, provided a large enough amount of search time

is given (but this is an important proviso).

3.1.2 Multi-stage GHH

The observations above provide the motivation for proposing a multi-stage GHH which

employs a different number of heuristics in two stages. The multi-stage GHH employs

“SCLR” in the first stage and “SCLxR” in the second stage to explore its search space as

much as possible (based on the heuristic list obtained in the first stage). The reason for

employing “SCLR” in the first stage of GHH is not only because it performs best, but also

because it employs the four distinct low level heuristics (while the LE, LD and LWD are


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based on the same heuristic and may contribute similarly to LD).

The results obtained for the multi-stage GHH are presented in Table 3, together with the

best results of the single-stage GHH in the last sub-section. We also present the state-of-the-

art approaches reported in the literature (Carter, Laporte and Lee, 1996, Di Gaspero and

Schaerf, 2000, Caramia, Dell’Olmo, and Italiano, 2001, Burke and Newall, 2002, Merlot et al,

2002, Asmuni, Abdullah et al, 2004, Burke and Garibaldi, 2004). They are reported as the

best results obtained by the corresponding approaches. We have not included an entry for the

corresponding computational time because they are not reported in several of these papers.


We can observe that the multi-stage GHH does not outperform the single-stage hyper-

heuristic presented in the last sub-section, although it obtains results that are not worse than

that of GHH upon “SCLR”. The reason may be that it starts from a smaller search space and

this may limit the search towards a certain region of the search space.

3.1.3 Summarizing Comments Concerning GHH for Exam Timetabling Problems

For all of the problems tested, GHH upon graph heuristics and random ordering obtained very

good results that are within the range of the best results reported in the literature. By

searching (on a higher level) the heuristic space, the GHH is capable of jumping (not moving)

among the solution space of the problem by moving among the heuristic lists in its search

space. We believe that this makes it a very efficient approach on difficult problems that have

complex, especially disjointed, solution spaces.

Also note that except for (Carter, Laporte and Lee, 1996) where a number of different

constructive methods with backtracking were employed, and (Asmuni, Burke and Garibaldi,

2004) where constructing, un-scheduling and rescheduling are performed, all of the other

approaches operate by improving on the initial complete solutions obtained beforehand. For


17

the complex problems tested here, it is observed that no feasible solutions can be obtained by

the use of pure constructive graph heuristics. In contrast, the proposed hyper-heuristic method

is constructive, starting from an empty solution. It is not dependent on initial solutions, which

may sometimes affect the behavior of the other approaches. Most importantly, it is also

effective for course timetabling problems (see below). Yet, the proposed method obtained

competitive results with all state-of-the-art approaches for exam timetabling problems.

We have found that deepest descent local search after each move of the Tabu Search

within the GHH approach often improves the solutions obtained and occasionally obtains a

better final solution than that obtained by the Tabu Search alone. This may indicate that,

during the problem solving, GHH sometimes reaches a point in the solution space (such as

solution a in Figure 2 and Figure 3) and cannot go further to its local optimal (such as solution

d in Figure 2 and Figure 3) unless moves upon the actual solutions are made. However, it is

difficult to check this hypothesis thoroughly, due to the size of the search space for the

solutions of complex timetabling problems. This is because the hypothesis concerns the

coverage of the solution space by the search space of the hyper-heuristic (see Section 2.2).

3.2 University Course Timetabling Problems

The proposed GHH method was also tested on eleven benchmark course timetabling

problems, proposed by the Metaheuristic Network1. This problem description is taken from

(Socha, Knowles and Sampels, 2002). The problems2 need to assign 100-400 courses into a

timetable of 5 days, 9 timeslots a day, while satisfying room features and capacity constraints.

They are grouped into small, medium and large problems. The hard constraints that must be

satisfied are:

1. no students can be scheduled to more than one event at a time

2. the room meets all features required by the event

1 http://www.metaheuristics.net/ 2 http://iridia.ulb.ac.be/~msampels/ttmn.data/


18

3. room capacity is respected

4. no more than one event is allowed per room and per timeslot

Soft constraint violations are equally weighted and summed up within the cost function. They

are presented below:

1. a student has a class in the last timeslot of the day

2. a student has more than two classes in a row

3. a student has only one class on a day

3.2.1 GHH upon All Low Level Heuristics

We employ exactly the same GHH approach with the same low level heuristics tested on the

exam timetabling problems. The only changes made, in order to deal with these different

problems, are the cost function which also evaluates the room constraints. The best results by

5 runs of GHH with distinct seeds upon all of the low level heuristics are presented in Table

4, which also presents the results of 3 approaches (Socha, Knowles and Sampels, 2002, and

Burke, Kendall and Soubeiga, 2003) reported in the literature for comparison. For the “Hyper-

heuristic” the best results out of 5 runs obtained are presented, for the “Local Search” and

“Ant Algorithm” the average results out of 50 runs on small problems, 40 runs on medium

problems and 10 runs on large problems are reported. We test our approach with the same

number of runs as that of the “Hyper-heuristic” from (Burke, Kendall and Soubeiga, 2003) to

make a more fair comparison. The term “x% Inf” in Table 4 indicates the percentage of runs

which failed to obtain feasible solutions.

From Table 4 we can observe that our GHH approach obtains competitive results with the

other 3 approaches on these course timetabling problems. For problem “Medium5”, it

obtained the best results among all approaches. It outperforms the “Local Search” method on

all of the problems except “Medium1”, “Medium2” and “Medium4”. And for all of the

problems tested it finds feasible solutions with all the 5 distinct seeds tested, which the “Tabu


19

Search Hyper-heuristic” and “Local Search” approaches failed to obtain.


3.2.2 Summarizing Comments Concerning GHH on Course Timetabling Problems

The experimental results on GHH for course timetabling problems demonstrate its ability to

work well on hard problems. We can observe that for problems “Medium5” and “Large”,

which are the hardest (highest costs obtained by the other approaches), GHH obtained the best

and second-best results. The reason may be that the GHH is capable of jumping (while not

moving) within the solution space by the moves within the search space of heuristic lists. This

enables it to deal well with a broad range of hard problems (with a complex, and probably

disjoint, solution space).

The deepest descent local search after each move of the Tabu Search improves the solution

in most cases during the search of GHH. The best final results are sometimes from the

improvement made by deepest descent local search on the solutions obtained by Tabu Search.

This may strengthen our hypothesis about the coverage of the solution space vs. search space

within our GHH approach.

For all the other approaches compared, initial solutions are required before the algorithms

are performed. Our GHH solves the problem by starting from an empty solution in each move

of Tabu Search. For the course timetabling problems tested here, the violations of soft

constraints 2 (more than 2 consecutive courses) and 3 (single course assigned in one day)

cannot be evaluated accurately until a complete solution is obtained. This is not true for the

other approaches compared here, since complete solutions exist. In our GHH approach the

evaluation can only be made approximately during the construction of the solution. This may

limit the search of GHH when searching for globally optimal solutions, although the deepest

descent local search method upon the complete solution after each step of Tabu Search can


20

improve the timetables concerning these soft constraints.

4. Conclusions

The overall goal of this paper was to investigate a hyper-heuristic which operates at a higher

level of generality than most of the current approaches studied in timetabling. Current state-

of-the-art approaches are the result of a significant investment of effort in developing

sophisticated and elaborate heuristics, which are “tailor made” for their particular problems.

In our hyper-heuristic framework both the Tabu Search and graph heuristics are general

methods that are widely applicable. We have presented a general constructive approach which

is not dependent on the initial solution that the other approaches need to generate.

By employing simple and general heuristics in an intelligent way, the hyper-heuristic is

capable of generating comparable results to those of special purpose approaches. The hyper-

heuristic has the ability of selecting general heuristics, picking up the events that are most

difficult to schedule (by different heuristics), in any given solution construction state.

Although the goal of the present study is not to beat the specific approaches in the literature,

the GHH works well on all of the problems and for one of the benchmark course timetabling

problems, the hyper-heuristic we developed actually obtained the best result among all those

reported in the literature. It is a simple, robust and very effective general approach that does

not use any domain knowledge except in the cost function that deals with different constraints

in different problems.

Experimental results indicate that the hyper-heuristic works more efficiently when using a

larger number of low level heuristics. However, the size of the search space of GHH increases

as well, which also increases the computational time. The multi-stage GHH was studied with

the aim of getting good results in the first stage of GHH employing less low level heuristics

and improving the results in the second stage employing more low level heuristics based on

the heuristic lists obtained from the first stage. However the experimental results presented

were worse than the single stage GHH due to the limitation of the starting points for the

search of GHH.


21

Our hypothesis on the coverage of the solution space of the problem by searching within

the search space of the hyper-heuristic is experimentally strengthened for both the exam and

course timetabling problems. This hypothesis needs to be tested on a larger range of

timetabling and scheduling problems.

5. Future Work

Future work could use other simple heuristics or approaches, instead of Tabu Search, in

exploring the search space of heuristics. A larger number of low level general heuristics can

also be added to the hyper-heuristic framework to explore a larger section of the solution

space of the problems. It might also be interesting to consider more than one low level

heuristic when choosing the events to be scheduled at each step of the solution construction.

Combining different low level heuristics at a single step of the solution construction may find

more appropriate events to be scheduled. Due to the large search space of the hyper-heuristic,

future work will also need to investigate the impact of additional low level heuristics on

computational time.

The hyper-heuristic approach described here represents a general and simple framework

equipped with little domain knowledge, and may be easily applied to many other timetabling

and scheduling problems with little effort. Future work should extend the same framework to

other problems.

Acknowledgements

We would like to acknowledge EPSRC for sponsorship of this research (grant number

GR/N36837/01). Professor Meisels’s contribution was supported by an EPSRC Visiting

Fellowship (GR/S53459/01). We would also like to thank the anonymous referees for their

helpful comments.


22

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Figure 1 Pseudo-code of Tabu Search within the graph based hyper-heuristic

initial heuristic list hl = {h1 h2 h3 … hk}

//Begin of Tabu Search

for i = 0 to i = (5 * the number of events) //number of iterations

h = change two heuristics in hl //a move in Tabu Search

if h does not match a heuristic list in ‘failed list’

if h is not in the tabu list //h is not recently visited

for j = 0 to j = k //h is used to construct a complete solution

schedule the first 2 events in the event list ordered using hj

if no feasible solution can be obtained

store h into the ‘failed list’ //update “failed list”

else if cost of solution c < the best cost cg obtained

save the best solution, cg = c //keep the best solution

add h into the tabu list

remove the first item from the tabu list if its length > 9

hl = h

//end if

Deepest descent on the complete solution obtained

//end of Tabu Search

output the best solution with cost of cg


29

Figure 2 The relation between the search space of the hyper-heuristic and solution space of the problem

search space of hyper-heuristic

A

B

C

unfeasible solutions

solution space of the problem

a

b

c

d

a neighbourhood move a “jump” move in the solution space

a correspondence between a heuristic list and the actual solution it generates


30

Figure 3 An illustration of how the deepest descent local search could lead to solutions that are not represented by

the heuristic lists (searched by the high level search)

search of hyper-heuristic leads (indirectly) to the following solutions (a to b and a to c)

b

a

c

d

the deepest descent then leads to solution d


31

Table 1 Characteristics of Benchmark Exam Timetabling Problems

car91 car92 ear83 hec92 kfu93 lse91 sta83 tre92 ute92 uta93 york83 exams 682 543 190 81 461 381 139 261 184 622 181

students 16925 18419 1125 2823 5349 2726 611 4360 2750 21266 941 timeslots 35 32 24 18 20 18 13 23 10 35 21

matrix density 0.13 0.14 0.27 0.42 0.6 0.6 0.14 0.18 0.8 0.13 0.29


32

Table 2 Costs of solutions obtained by GHH upon a different number of heuristics (S: saturation degree,

L: largest degree, C: color degree; R: random ordering, Lx: largest weighted, largest enrollment and

largest degree)

Problem SL SLR SCL SCLR SCLx SCLxR SCLxR(10*) car91 5.73 5.55 5.51 5.41 5.78 6.13 5.67 car92 5.01 4.79 4.89 4.84 4.88 4.93 4.91 ear83 40.54 39.1 40.47 39.17 38.19 40.37 40.23 hec92 13.41 12.86 13.03 13.11 13.89 12.72 12.55 kfu93 16.63 15.83 15.76 16.01 15.91 17.03 15.83 lse91 13.71 13.88 14.12 13.46 13.15 13.88 13.11 sta83 150.22 143.3 148.54 143.75 141.08 142.83 142.03 tre92 9.19 9.38 8.85 9.27 8.97 9.27 9.15 ute92 32.71 32.01 32.03 32.01 31.65 32.1 31.83 uta93 4.07 4.18 4.0 3.54 3.75 3.77 3.65

york83 46.21 44.0 45.05 44.51 40.13 48.78 46.03


33

Table 3 Results from the GHH, multi-stage GHH and the best results reported in literature on benchmark exam

timetabling problems

car91 car92 ear83 hec92 kfu93 lse91 sta83 tre92 ute92 uta93 york83 GHH (best) 5.41 4.84 38.19 12.72 15.76 13.15 141.08 8.85 32.01 3.54 40.13 Multi-stage

GHH 5.41 4.84 38.84 13.11 15.99 13.43 142.19 9.2 31.65 3.54 44.51

Abdullah et al 5.21 4.36 34.87 10.28 13.46 10.24 150.28 8.13 24.21 3.63 36.11 Asmuni et al 5.20 4.52 37.02 11.78 15.81 12.09 160.42 8.67 27.78 3.57 40.66

Burke &Newall 2002

4.6 4.0 37.05 11.54 13.9 10.82 168.73 8.35 25.83 3.2 36.8

Burke,Bykov et al 2004

4.2 4.8 35.4 10.8 13.7 10.4 159.1 8.3 25.7 3.4 36.7

Caramia et al 6.6 6.0 29.3 9.2 13.8 9.6 158.2 9.4 24.4 3.5 36.2 Casey &

Thompson 5.4 4.4 34.8 10.8 14.1 14.7 134.7 8.7 25.4 Inf 37.5

Carter et al 7.1 6.2 36.4 10.8 14.0 10.5 161.5 9.6 25.8 3.5 41.7 Di Gapero & Schaerf

6.2 5.2 45.7 12.4 18.0 15.5 160.8 10.0 29.0 4.2 41.0

Merlot et al 5.1 4.3 35.1 10.6 13.5 10.5 157.3 8.4 25.1 3.5 37.4


34

Table 4 Results of the GHH, multi-stage GHH and the other 3 approaches in literature on benchmark course

timetabling problems

GHH

upon 6 heuristics

Tabu Search Hyper-Heuristic Burke, Kendall

&Soubeiga 2003

Local Search Socha, Knowles & Sampels 2002

Ant Algorithm Socha, Knowles & Sampels 2002

Small1 6 1 8 1 Small2 7 2 11 3 Small3 3 0 8 1 Small4 3 1 7 1 Small5 4 0 5 0

Medium1 372 146 199 195 Medium2 419 173 202.5 184 Medium3 359 267 77.5% Inf 248 Medium4 348 169 177.5 164.5 Medium5 171 303 100% Inf 219.5

Large 1068 80% Inf 1166 100% Inf 851.5

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